$ A = \left[\begin{array}{rr}2 & 1 \\ -1 & 2\end{array}\right]$ $ D = \left[\begin{array}{rrr}5 & -2 & -2 \\ -2 & 2 & 3\end{array}\right]$ What is $ A D$ ?
Solution: Because $ A$ has dimensions $(2\times2)$ and $ D$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ A D = \left[\begin{array}{rr}{2} & {1} \\ {-1} & {2}\end{array}\right] \left[\begin{array}{rrr}{5} & \color{#DF0030}{-2} & \color{#9D38BD}{-2} \\ {-2} & \color{#DF0030}{2} & \color{#9D38BD}{3}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{2}\cdot{5}+{1}\cdot{-2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{5}+{1}\cdot{-2} & ? & ? \\ {-1}\cdot{5}+{2}\cdot{-2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{5}+{1}\cdot{-2} & {2}\cdot\color{#DF0030}{-2}+{1}\cdot\color{#DF0030}{2} & ? \\ {-1}\cdot{5}+{2}\cdot{-2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{2}\cdot{5}+{1}\cdot{-2} & {2}\cdot\color{#DF0030}{-2}+{1}\cdot\color{#DF0030}{2} & {2}\cdot\color{#9D38BD}{-2}+{1}\cdot\color{#9D38BD}{3} \\ {-1}\cdot{5}+{2}\cdot{-2} & {-1}\cdot\color{#DF0030}{-2}+{2}\cdot\color{#DF0030}{2} & {-1}\cdot\color{#9D38BD}{-2}+{2}\cdot\color{#9D38BD}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}8 & -2 & -1 \\ -9 & 6 & 8\end{array}\right] $